Hemispherical drive for autonomic and unconstrained actuation of flapping fins with variable amplitude

ABSTRACT

A small hemispherical drive apparatus actuates the cilia to undergo the limit cycle oscillations. Electromagnets are positioned on a housing of the apparatus. A gimbal with shaft is also attached to the housing. The shaft has a first end within the housing that is proximally separated and remains separated from the electromagnets as the shaft rotates. Generated signals excite electromagnets in sequence to produce an electromagnetic track for the shaft.

This application is a divisional application and claims the benefit ofU.S. patent application Ser. No. 14/164,360 filed on Jan. 27, 2014 whichclaims the benefit of U.S. Provisional Patent Application No.61/849,912; filed on Jan. 28, 2013 by the inventor, Dr. PromodeBandyopadhyay and entitled “ACOUSTO-OPTICAL METHOD OF ENCODING ANDVISUALIZATION OF UNDERWATER SPACE”.

CROSS REFERENCE TO OTHER PATENT APPLICATIONS

None.

STATEMENT OF GOVERNMENT INTEREST

The invention described herein may be manufactured and used by or forthe Government of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefor.

BACKGROUND OF THE INVENTION

(1) Field of the Invention

The present invention relates to an autonomic based method of nonlinearphase encoding and hologram-based transmission of coded echoes from anunstructured underwater environment with real time decoding andvisualization of the environment.

(2) Description of the Prior Art

Visualizing underwater space, particularly in three-dimensions and inreal time is of utmost importance. This visualization is accomplishedmostly by passive or active sonar devices. Advanced arrays have beendeveloped with serious shortcomings. Therefore, alternative concepts,theories and hardware are needed that would lead to a fundamentallydifferent approach to visualizing underwater space.

In the natural world, large swimming animals such as dolphins inhabitshallow waters, and whales inhabit the deeper ocean. These separatespecies communicate with each other in each species and accuratelydetect objects. There is evidence that dolphins create an image of theenvironment using sonar in which the image is similar to the visiblepart of the electromagnetic spectrum. There does not seem to be muchdifference in the manner of detection between nearly blinded dolphinswhen in turbid rivers and the dolphins in oceans where there is morevisibility.

Co-directivity is describable to justify a nonlinear approach toacoustic encoding of the environment. It is assumed that animals have toremain in synchrony with the environment to survive. It is also assumedthat for persistent synchrony with the environment; the sensors, motioncontrol neurons and the actuators need to have the same dynamics—namelynonlinear oscillatory autonomic. In other words, self-correctingdynamics of the Na and Ca ions in the membranes are closely related tothe sensing and the flopping of motion actuators.

Neurons evolved to allow movement of life forms from point to point. Thebrain became more complicated to fulfill the needs of movements that hadto be more complicated. Complexities in environments require a greatdeal of sensing which creates a difficult problem in the design ofautonomous platforms. The question of should there be any rationalfoundation of integrating sensors, controllers and actuators in aplatform is considered.

Observations made in the hearing and swimming propulsion of animalssuggest that animals have common dynamics probably determined by anintervening controller. Simulations are subsequently carried out whichshow that benefits in homing may accrue from such common dynamics whichexhibits a preferred ‘handedness’. It is hypothesized that sensors,controllers and actuators have common autonomic oscillatory nonlineardynamics. This allows animals to be in persistent oscillatory synchronywith the environment.

The dynamics of the olivo-cerebellar neuron is measured using an analogcircuit. Features of observed trajectories of bats and of cilium ofparamecium are calculated using the olivo-cerebellar dynamics and it issuggested that chaos helps a platform to adapt to changes inenvironment.

To understand how animals achieve such feats; is there any commonfoundation in their sensing, control and propulsion? Animals can have alarge number of sensors in their body for mapping the environment andsensing changes. In manmade platforms, an increase in sensing isdemanding on processing and coordination with the controller and themotion actuators. Therefore, it is reasonable to expect that a commonfoundation in sensing, control and the mechanism of propulsion could bethe key to autonomy. In this work, there is evidence for the existenceof such a common foundation, and carry out simulations of motion todetermine advantages that such a foundation might offer. Finally, thenotional design of a nonlinear volumetric and metachromic sensor ispossible in which the design could offer autonomy in an unstructuredenvironment that seem to be lacking in man-made platforms of today.

It is important to understand that muscles of animals are vibrating atroughly 10 Hz, however imperceptible the amplitude at a given instantmay be. This vibration is not monochromatic and is nonlinear. Note thatthe motion of animals is controlled by inferior-olive neurons which aremathematically described as coupled nonlinear oscillators that areslightly unstable. They have a property called ‘Self-Referential PhaseReset’ whereby an external impulse can bring any number of de-correlatedactuators (muscles) into a common phase.

It has been proposed that persistent synchrony with the environment, inall animals; the motion control inferior-olive neurons, the actuators(muscles) and the sensors operate on the same nonlinear dynamical systemprinciples. Using this assumption and some principles of handedness; itwas shown that a platform/animal would be able to home on to a movingtarget faster.

In order to enhance detection; a need therefore exists for an accessibleversion of the visualization process presently used in the naturalworld.

SUMMARY OF THE INVENTION

Accordingly, it is a general purpose and primary object of the presentinvention to provide a method for visualizing underwater space.

It is a further object of the present invention to provide a method fornonlinear phase encoding and hologram-based transmission of coded echoesfrom an underwater environment.

It is a still further object of the present invention to provide ahemispherical drive apparatus that provides cilium actuation receptiveto nonlinear phase encoding.

To attain the objects described, the term “cilium” is used to define atransducer that is non-linear with an oscillating flexible line in whichthe flexible line is anchored at one end. The dynamics of a cilium isprovided by a second order non-linear ordinary differential equation.Normally, the cilium vibrates in a track that is described as a “limitcycle”. In other words, the cilium has self-correcting dynamics. Thisself-correcting control is termed “autonomic”.

It has been found that the motion of a cilium can be calculated bysumming the orthogonal oscillations of the cilium. In the presentinvention, the torques are applied by a hemispherical drive apparatus tothe cilium with the apparatus providing a limit cycle oscillation (LCO).When a perturbation is applied to a cilium; the cilium will be disturbedfrom the limit cycle track.

During operation, reference coding and an echo coding of a target iscreated; a hologram is transmitted; and the hologram is decoded at aremote point. The reference coding includes acoustic illumination thatdisturbs the limit cycle oscillation of the cilium. This disturbed limitcycle oscillation is recorded as a reference hologram. Acousticillumination upon the target is echoed to a dense cluster of the ciliaon which the echo is encoded. A temporal wave amplitude, direction andfrequency content get encoded into the cilium position and posture. Thetrack and speed with which the cilium returns to the limit cycleoscillation (the autonomic properties of the cilium); providesinformation on an input disturbance to the cilia.

Metachronism is used to read input to the cilium. A wave motion of agroup of cilia is called a metachronic wave. The metachronic wave is afootprint of a large scale external impulse created by the motion ofindividual vibrating elements.

Also, each cilium is time dependent. Being time dependent provides theability to encode a large amount of amplitude and phase information.Restated, the cilium motion is a limit cycle oscillation. If an inputdisturbance dislodges the cilium motion of the limit cycle from point Ato B; the autonomic properties will return the cilium to the limit cycleoscillation after the disturbance has passed. The disturbance propertiessuch as strength, direction, and frequency plus the arrival timedetermines the location of A and B. The path and speed that the ciliumtakes from B to return to the limit cycle oscillation are provided byautonomic equations. The equations also provide the required inputdisturbance properties.

The individual cilia are also sensitive to the direction and amplitudeof the wave. If the wave is nonlinear and has a phase identity; then thecollection of cilia will produce a visual image of the imposeddisturbance. If there is a reference motion due to a transmitted wave,and the later imposed wave is an echo; then by differencing with thetransmitted wave, an image of the surroundings from which the echooriginates can be created.

In a multi-cilia configuration, when a neighboring cilia beats with aconstant phase difference; the overall wave pattern is termedmetachronic. For encoding the acoustically-painted object; a metachronicwave pattern would have to be created using the multi-ciliaconfiguration. The echo distorts the autonomous reference metachronicwave pattern. This distortion is optically measured via the clusteredpattern of the changes in the curvature and the torsion of individualcilia.

The system of the present invention also includes a hemispherical driveapparatus for cilium actuation in which the apparatus includes aplurality of electromagnets with each electromagnet having correspondinglatitude and longitude coordinates.

The drive apparatus further comprises a gimbal positioned in a housingof the apparatus, and a shaft attached to the gimbal. The gimbal allowsthe shaft to roll and oscillate in different planes, wherein each planeis orthogonal to the other planes. The shaft remains spaced apart fromthe electromagnetic array by a predetermined distance as the shaft rolland oscillates. A fin is attached to an end portion of the shaft spacedapart from the housing.

The drive apparatus further comprises a control system with a controllerfor storing the coordinates of each electromagnet. The control systemalso includes signal generation circuitry for outputting signals thatexcite electromagnets in a predetermined order. The excitation of theelectromagnets produces an electromagnetic track having flux lines thatflow through the shaft; thereby, causing the fin to flap.

The control system also includes a computer for programming thecontroller with at least one sequence of latitude and longitudecoordinates of electromagnets that are to be sequentially excited orderin order to form the electromagnetic track. One pair of adjacentelectromagnets is excited at a time and each succeeding pair of excitedelectromagnets is adjacent to a pair of previously excitedelectromagnets.

The control system drives a fin or cilium. For use of the flapping fin;the advantages of the hemispherical drive apparatus over conventionalshaft and gear drives are: common oscillating shafts have nonuniformwear, hence there is leakage; the shafts each require too many gears;the shaft drives have vibration, backlash and are noisy; and the shaftdrives are cumbersome in the actuation of roll and pitch.

BRIEF DESCRIPTION OF THE DRAWINGS

Other objects, features and advantages of the present invention willbecome apparent upon reference to the following description of thepreferred embodiments and to the drawings, wherein correspondingreference characters indicate corresponding parts throughout the severalviews of the drawings and wherein:

FIG. 1 is a schematic depicting orthogonal oscillations of a cilium;

FIG. 2A is a representation of an acoustic illumination of a target;

FIG. 2B is a representation of echo coding of a target;

FIG. 3 is a flow chart representing the collection of a referenceacoustic wave packet;

FIG. 4 is a schematic depicting creation of a hologram of athree-dimensional environment;

FIG. 5 is a schematic of a hologram reconstruction of a virtual object;

FIG. 6 is a screen depiction of an animation for an array of ciliumtransducer elements on which successive point blasts from acousticallypainted sources are imprinted;

FIG. 7 is a van der Pol simulation of a typical limit cycle in which thesimulation depicts cases of damped (inside the limit cycle) andamplified (outside the limit cycle) conditions;

FIG. 8A is a position plot comparison of limit cycle oscillation (LCO)tracks of the cilium of a paramecium with computational tracks of theoscillator;

FIG. 8B is a velocity plot comparison of the limit cycle oscillation(LCO) tracks of the cilium of a paramecium with computational tracks ofthe oscillator;

FIG. 9A is a plot depicting the relationship between two nonlinearoscillators with respect to time traces of the variables constantamplitude (w1) and dropping amplitude (z2);

FIG. 9B is a plot depicting state orbits of the oscillators of FIG. 9Aplotted between times marked by the vertical lines of FIG. 9A;

FIG. 10 is a isometric view of a hemispherical drive of the presentinvention in which the drive provides orthogonal roll motions to acilium;

FIG. 11 is a diagram depicting the position and movement of a shaft andfin with respect to an array of electromagnets positioned in ahemispherical housing of the present invention;

FIG. 12 is a diagram illustrating the distance or spacing between theelectromagnets and the first end of the shaft as a gimbal (depicted inFIG. 10) allowing the shaft to roll and oscillate in different planes;

FIG. 13 is a diagram of the shaft and fin attached thereto;

FIG. 14 is a front view of an electromagnet of the hemispherical driveapparatus of the present invention; and

FIG. 15 is a block diagram of a control system for generating signalsthat excite predetermined electromagnets in order to produce anelectromagnetic track.

DETAILED DESCRIPTION OF THE INVENTION

In the description, the term “cilium” is used to define an individualtransducer. A cilium is non-linear with a three-dimensional oscillatingflexible line in which the flexible line is anchored at one end. Thedynamics of a cilium is provided by a second order non-linear ordinarydifferential equation. Normally, the cilium vibrates in a track that isdescribed as a “limit cycle”. In other words, the cilium hasself-correcting dynamics.

If the vibration is damped; the cilium is naturally amplified to returnthe motion to the limit cycle. Alternatively, if the vibration isamplified, the cilium is naturally suppressed; whereby, the vibrationreturns to the limit cycle. This self-correcting sensor-less control istermed “autonomic”.

Referring now to FIG. 1; it has been found that the motion of a ciliumcan be created by summing the orthogonal oscillations of the cilium. Asper aero-elasticity theory (in certain frequency ranges and depending onthe viscosity of the surrounding fluid); the motions can couple andthere can be an exchange of energy between the motions. For an animal(such as paramecium), these torques are applied by the animal.

In the present invention, such torques are applied as a limit cycleoscillation (LCO) by a hemispherical drive apparatus 10 (see FIG. 10).When a perturbation is applied to a cilium; the cilium will be disturbedfrom the limit cycle track. The perturbation tensor is encoded in a mapof a disturbed state space. The amplitude and phase of each cilium canbe tensorially represented. The tensor is encoded into the temporalposition, velocity and acceleration of the cilium multitude.

Generally, a target (object) is coded; a hologram is transmitted; andthe hologram is decoded at a remote point. The target exists inunstructured temporal space and is coded with an imprint of thethree-dimensional space representing numerous elements. Thethree-dimensional space can be separated into numerous mirrors slantedat different orientations from which an incident sound returns to thecilium. The echo from each mirror will include information on range,orientation and softness/hardness (material) composition. The imprint istransmitted; decoded; and a virtual object is created.

FIG. 2A and FIG. 2B depict the steps of creating a reference coding andan echo coding of the target 100. The reference coding includes acousticillumination that disturbs the limit cycle oscillation of the cilium.This disturbed limit cycle oscillation is recorded as a referencehologram. Acoustic illumination upon the target 100 is also echoed to adense cluster of the cilia on which the echo is encoded. It should benoted that two holograms can be created to capture the reference and theecho. These holograms can then be transmitted over distances forprocessing.

FIG. 3 depicts a flow chart 70 of steps for collection of a referenceacoustic wave packet and surroundings on a three-dimensional array byacoustic painting and encoding of an environment in a cluster of ciliumtransducers. In step 72, a reference acoustic wave packet is definedusing existing collections of gathered wave packets and broadcast instep 74. The target (object) 100 is acoustically painted in step 76 andan echo is received in step 78. The echo is codified in step 80 with theecho disturbing the cilia. The location of track dislocation (A-B inFIG. 7) and the track taken by the cilium to return to the limit cycleoscillation contain information on the incident disturbance (includingthe sound wave). The parameters are the positions of A, B, and C (thepoint of return) shown in FIG. 7. The track position, velocity andacceleration of paths in the figure would be converted to pressure waveinformation using calibration and known nonlinear equations for thecilium motion.

Returning to FIG. 3 and in step 82, a metachronic three-dimensionalpattern is reflected in the reference and echo hologram and in step 84,the two metachronic patterns are converted into two holograms (a videoof each cilium motion and their grouping is converted into a hologram).In step 86, the holograms are transmitted to a remote location and instep 88; the holograms are formulated and displayed. In step 90, theobject (target 100) is decoded by differentiating the reference hologramfrom the echo hologram (differentiation is achievable by existingtechnology). Lastly in step 92; the virtual object (representing thetarget 100) is displayed.

Unlike a monochromatic frequency sweep of conventional sonar; in thepresent invention, the object/target is painted by a defined nonlinearacoustic wave packet. Due to the delayed arrival of the echo; thereference wave packet acts to start the limit cycle on the cilium 12.The echo arrives and disturbs the trajectories of the cilium 12. Minimalenergy input (emission of one wave packet) is needed in the environmentin active mode and continuous emission is not required.

Due to the nonlinear wave packet nature of the ‘paint’; the coherence inthe littoral (reverberating) areas is not lost. In explanation bynatural world use, dolphins use chirps (wave packets) to ping objects inshallow water littoral regions and the wave packets do not lose theircoherence. In signal processing using Fourier decomposition;relationships between frequencies are not easy to compute and are notfrequently considered. In the natural world, chirps (wave packets)retain an identity over a distance because the wave packet consists of aspecific group of frequencies that are related thru the same mechanism(equations) with specific amplitudes, dominant frequency content andphase relationship. The underlying mechanism is common toself-regulation.

FIG. 4 depicts how a hologram of the imprint of a three-dimensionalenvironment is created. In the figure, a pulsed laser (coherent lightbeam) 200 is split with a beam splitter 202 with a portion of the beamforming an acoustic illumination for the reference beam and a portion ofthe beam forming an acoustic illumination of the target 100.

FIG. 5 shows a hologram reconstruction of a virtual object beingtransmitted to a recording plate 300 at a remote facility. Two hologramsare reconstructed at the remote facility—one hologram of the effects ofthe reference wave packet and one hologram due to the echo off thetarget 100.

The reference acoustic wave packet can be dolphin-inspired. A library ofdolphin pings can be used as the diagnostic wave packet in the presentinvention. Such wave packets have a range of frequencies and maintaincoherence in the reverberating environment of the shallow waters of alittoral ocean.

Typically, wave packets appear in a burst. However, on inspection, thereare time gaps between sub-bursts during which the echo is received. Thebursts are tuned and altered based on the echo. These pings arecalibrated for the boundary conditions of the tasks. The dolphin pinglibrary can be used for both the reference and the echo generation.

Described below is a theory of operation of the present invention. Theoperation is not meant to limit the present invention but insteadprovides a theory as to the usefulness of the invention.

FIG. 6 depicts a simulation of how a 5×5×5 cluster of cilia wouldrespond to two spherically expanding sounds originating at the lowerright corner near the reader; followed by another blast from the middleof the vertical left face (marked as 1 and 2, respectively). The frontof the second blast is shown as a sphere with the edge beingrecognizable. Time is represented by dark shading in which the darknessincreases as the shading increases. The time is denoted by light shadingwhen the cilia are set to the reference vibration. Subsequently, afterthe target 100 is subjected to a sound wave; if the target faces anotherwave from any direction then the light shading resumes. Each cilium 10traverses a nonlinear limit cycle described by a van der Pol oscillatorgiven by Equation (1). The lightest shaded lines at each cilium base arereference orientations.

FIG. 7 depicts a limit cycle for parameter values in Equation (1).Suppose the state is at “A” when the echo arrives to dislodge the cilium10 to “B”. The system will then return to the limit cycle near “C”.Therefore, the relative location of “B” with respect to “A” and thereturn path to “C” capture the nonlinear information contained in theecho.

The map in FIG. 7 is two-dimensional; however, three-dimensional mapscan be created using the states (z, v, w)—see Equation (1). The speedand path that the system takes to return to the limit cycle (“B” to “C”)after being disturbed by the echo (“A” to “B”) can be described byvectors (u(x,t)). Although pressure is scalar; the dynamics of the soundwaves has been encoded using tensor. The conversion of echo pressuredata to cilium position data is an encoding. It should be noted that alinearly vibrating rod does not have the ability to capture the amountof phase information as would a nonlinearly vibrating element.

Equation (1) shows the description of the inferior-olive neuron dynamicsthat controls the motion of animals. This description shows two couplednonlinear oscillators; the nonlinear terms are given by “p”. Theoscillator is given by second order ordinary differential equations.

$\begin{matrix}{{\begin{bmatrix}{\overset{\bullet}{u}}_{i} \\\begin{matrix}{\overset{\bullet}{v}}_{i} \\\begin{matrix}{\overset{\bullet}{z}}_{i} \\{\overset{\bullet}{w}}_{i}\end{matrix}\end{matrix}\end{bmatrix} = {\begin{bmatrix}{k\;{ɛ_{N\;\alpha}^{- 1}\left( {{p_{iu}\left( u_{i} \right)} - v_{i}} \right)}} \\{k\left( {u_{i} - z_{i} + I_{C\;\alpha} - I_{N\;\alpha}} \right)} \\{{p_{iz}\left( z_{i} \right)} - w_{i}} \\{ɛ_{C\;\alpha}\left( {z_{i} - I_{C\;\alpha}} \right)}\end{bmatrix} + {\begin{bmatrix}0 \\0 \\0 \\{- ɛ_{C\;\alpha}}\end{bmatrix}{I_{exti}(t)}}}}\begin{matrix}{{p_{iu}\left( u_{i} \right)} = {{u_{i}\left( {u_{i} - a} \right)}\left( {1 - u_{i}} \right)}} \\{{p_{iz}\left( z_{i} \right)} = {{z_{i}\left( {z_{i} - a_{i}} \right)}{\left( {1 - z_{i}} \right).}}}\end{matrix}} & (1)\end{matrix}$

The upper oscillator, which has a spikey and rich dynamic, is notneeded. Instead, the transducer motion is modeled as a nonlinearoscillator where, for the ith oscillator, the states z_(i) and w_(i) aregiven by the differential Equation (2).ż _(i) =p _(iz)(z _(i))−w _(i){dot over (w)} _(i)=ε(z _(i) −I)  (2)where the nonlinear function is given byp_(iz)(z_(i))=z_(i)(z_(i)−a_(i))(1−z_(i)), ε is a constant parametercontrolling the time scale, a_(i) is a constant parameter associatedwith the nonlinear function and I is a constant parameter. Equation (2)can be written as{umlaut over (z)} ₁ +F(z _(i))ż _(i) +kz _(i) +εI=0  (3)where F is a cubic polynomial function and k is a constant. Equation (3)resembles Lienard's oscillator (in contrast, the function F is awell-defined quadratic in the familiar van der Pol's oscillator). Theoscillator exhibits a closed orbit in the state space that is(z_(i)−w_(i)), which is also known as limit cycle oscillation with theconstant parameters determining the form of the closed orbit.

FIG. 8A and FIG. 8B show position and velocity tracks, respectively;comparing the limit cycle oscillation tracks of the cilium of aparamecium and the computation using Lienard's oscillator as perEquation (2). On both figures, the solid line is an equation model andthe broken line shown is the actual measurement of the cilium positionand velocity—which is the analog of the transducer. As seen, a closecorrelation is obtained. The simulation shows that the cilium follows atrack—described as a limit cycle. Therefore, the cilium has an autonomiccharacter wherein the cilium is controlled without sensors. Thenonlinear autonomic nature of the dynamics of the cilium ensures thatthe phase coding of the echo is robust.

A multiplicity of nonlinear sensors captures additional relative phaseinformation. One example considers two sensors, where i=1 and 2 inEquation (2). The temporal relationship between the two oscillators isshown in FIG. 9A and FIG. 9B. Note that while limit cycle oscillationexists in each oscillator; the relationship between the two oscillatorsis not resonant and the relationship is changing. The incident nonlinearacoustic wave packets (the reference or the echo) would affect eachtransducer differently and would also have a joint spatially-distributedeffect.

The nonlinear interactive approach would be able to capture the effect.This is an example of phase information that a row of linearlyoscillating transducers would not be able to capture. The nonlinearoscillators in Equation (2) are self-correcting; therefore, theoscillators robustly capture the phase information.

The wave motion of a group of cilia is called a metachronic wave. Themetachronic wave is a collective footprint of a large scale externalimpulse created by the motion of individual vibrating elements.Therefore, individual cilia are time dependent and are sensitive to thedirection and amplitude of the wave. If the wave is nonlinear and is apacket of waves that has a phase identity; then the collection of ciliawill produce a visual image of the imposed disturbance. If there is areference motion due to a transmitted wave, and the later imposed waveis an echo; then by differencing with the transmitted wave, an image ofthe surroundings from which the echo is coming can be created. This canbe described by the difference set or tensor. In a planar situation, anorthogonal axis system exists wherein along one of the axes; the ciliawill be in phase and orthogonal to the plane where the cilia are inphase; there will be a phase difference between the neighboring cilia.The spacing between the neighboring cilia has to be of a certain valuefor the metachronic wave to develop autonomously. Spacing depends on thecilium length and fluid viscosity in which the cilium are submerged andthe rotational frequencies of the cilium.

The nonlinear dynamics of the metachromic wave pattern is robust due toautonomous control of the wave pattern. This is an analog solution andtherefore is comparatively fast and less computer intensive. This is animportant departure from current theories of sonar transducers which arebasically linear and digital. The current approach is similar to aclosed form solution. Displacement accuracy of the transducer is on theorder of Angstroms. Because electronics is not involved; frequencyresolution is high.

FIG. 10 depicts the hemispherical drive apparatus 10 that providesorthogonal roll motions to the cilium 12. In the figure; thehemispherical drive apparatus 10 has a housing 16 which has an interior18 and a centered tip 20. The hemispherical drive apparatus 10 furthercomprises a plurality of electromagnets 22 positioned within theinterior 18 and arranged in an array. The location of each electromagnet22 is defined by latitude and longitude coordinates. Fabrication usingelectrical engineering may use an (i, j) matrix to position theelectromagnets. Analytically, noise engineers would prefer a polarsystem (phi, theta). In one embodiment, there are thirty-fiveelectromagnets 22.

The hemispherical housing 16 includes a cover 24 for the interior 18.The hemispherical housing 16 also includes a waterproof membrane 26which is disposed over the cover 24 (See FIG. 11). The waterproofmembrane 26 has a side 28 which contacts the cover 24 and a side 30which is exposed to water (seawater).

The hemispherical drive apparatus 10 further comprises a gimbal 32 thatis attached to the hemispherical housing 16 and located on thewaterproof membrane 26. The hemispherical drive apparatus 10 alsocomprises a shaft 34 that is attached to gimbal 32 such that the shaftis able to roll and oscillate in different planes wherein one plane orrotational axis is orthogonal to the other plane or rotational axis.

In FIG. 11, the shaft 34 has a first end portion 36 which comprisesferromagnetic material and is located within the interior 18 of thehemispherical housing 16. In one embodiment, the first end 36 isconfigured as a rounded section. The shaft 34 further comprises a secondend portion 38 which is external to the interior 18. The first end 36 isspaced apart from the array of electromagnets 22 by a predetermineddistance D. The first end 36 remains spaced apart at the predetermineddistance D as the shaft 34 rotates in roll oscillation planes φ1 and φ2(See FIG. 10). The first end 36 does not contact the electromagnets 22.

As shown in FIG. 12, the shaft 34 has a hollow interior and an internalshaft 40 located within the interior. The internal shaft 40 is free torotate within the hollow interior of the shaft 34 about pitch axis θ. Aninternal pitch motor (not shown) is located within the hollow interiorof the shaft 34 and rotates the internal shaft 40 about the pitch axisθ. The internal pitch motor may receive control signals from a controlsystem 60 shown in FIG. 15.

Returning to FIG. 12; the hemispherical drive apparatus 10 furthercomprises a propulsion fin 42 that is attached to a distal end 44 of theinternal shaft 40. The drive apparatus 10 can be built in 10 centimeter(large) scale for flapping fin propulsion as well as 100 micron (small)scale for cilium actuation at a small scale. The cilium length may be 17microns with a diameter of 1 micron. Both flapping fins and ciliummotion follow known nonlinear autonomic equations. Both flapping finsand cilium reject disturbances autonomically (without the need of anysensor or controller) and are robust as a result.

The fin 42 functions as a flapping fin with a leading edge 46 andtrailing edge 48. The fin 42 is attached at the ⅓ chord point at the endof the internal shaft 40. Measured from the leading edge 46, the fin 42is hinged for pitch at this distance of “c/3” where “c” is the finchord. Pitching at c/3 maximizes hydro efficiency.

As the internal shaft 40 rotates about the pitch axis θ; the flappingfin 42 also rotates about the pitch axis θ. Since pitch oscillationoccurs internally to the shaft 34; there is no leakage. A first plane ofoscillation is roll oscillation φ1 and a second plane of oscillation isroll oscillation φ2. Roll oscillation plane φ2 is orthogonal to rolloscillation plane φ1.

The viability of the hemispherical drive apparatus 10 can be estimatedby calculating the magnetomotive force F, as:

$\begin{matrix}{F = \frac{({ni})^{2}\mu_{0}A}{2l_{gap}^{2}}} & (4)\end{matrix}$where n=number of turns in the coil, i=current through coil (A),μ₀=magnetic permeability in a vacuum, A=area; and l_(gap)=air gapbetween the magnet and steel in the direction of magnetic flux. Comparedto μ₀, magnetic permeability of cold rolled steel is 2,000 times larger.For a gap, l_(gap) of 3 to 4 mm, at i=1.5 A, both Equation (4) andmeasurement of maximum force lead to a force of 0.75 N.

Referring to FIG. 15, the control system 60 generates signals thatsequentially excite the predetermined electromagnets 22 in accordancewith a programmed sequential order. Exciting the predeterminedelectromagnets 22 produces an electromagnetic track. An example of anelectromagnetic track 400 is shown in FIG. 10. The electromagnetic track400 is in the form of a figure-eight pattern or a curve. It is to beunderstood that the center of the electromagnetic track 400 does notneed to be at the center of the hemispherical housing 16. The length ofthe shaft 34 in centimeters determines the scale in millimeters of thefigure-eight electromagnet track 400. Since the first end 36 of theshaft 34 comprises ferromagnetic material; the flux lines created by theexcited electromagnets 22 pass through the ferromagnetic material. As aresult, the first end 36 moves along the electromagnetic track 400. Dueto the function of the gimbal 32; the movement of the first end 36causes movement of the fin 42.

Referring again to FIG. 15, the control system 60 comprises a controller62. The controller 62 has a programmable processor that stores thelatitude and longitude coordinates of each electromagnet 22 and aplurality of sequences of latitude and longitude coordinates of theelectromagnets that are to be excited in sequential order in order toproduce the electromagnetic track 400. Each sequence of latitude andlongitude coordinates of the electromagnets 22 comprises a sequence oflatitude and longitude coordinates of pairs of adjacent electromagnets.One pair of adjacent electromagnets 22 is excited at a time and eachsucceeding pair of excited electromagnets is adjacent to a pair ofpreviously excited electromagnets. In such an embodiment, theelectromagnetic track 400 lies between the electromagnets 22 of eachpair of adjacent electromagnets.

The controller 62 outputs digital signals that contain data thatcorrespond to the latitude and longitude coordinates of theelectromagnets 22 that are to be excited. The control system 60 includesinput/output circuitry 64 that comprises an array of six analog circuitsthat solve the second ordinary nonlinear ordinary differential equationsgoverning the self-regulating process which is represented by Equation(5) in the ensuing description.

The input/output circuitry 64 outputs signals that are inputted into adigital-to-analog converter (DAQ) 66. The DAQ 66 converts the signals toanalog signals that are inputted into the array of electromagnets 22 ofthe hemispherical drive apparatus 10. A status signal is outputted bythe array of electromagnets 22 and inputted into the DAQ 66.

The control system 60 further comprises a computer 68. The computer 68is used to program the controller 62 and is also in signal communicationwith the DAQ 66 via a data bus 69. The computer 68 generates DAQ controlsignals for input into the DAQ 66. The computer 68 programs theprogrammable processor of controller 62 with the plurality of differentsequences of latitude and longitude coordinates of the electromagnets 22that are to be excited in sequential order.

The control system 60 allows the center of the electromagnetic track 400to be displaced to a desired position that is not at the center of thehemispherical housing 16. The control system 60 also allows fordistortion of the curve of the electromagnetic track 400. This featureallows formation of an electromagnetic track that follows a pattern orcurve other than a figure eight.

The fin 42 can be flapped in a desired orientation by using the controlsystem 60 to excite certain electromagnets 22 to produce anelectromagnetic track that vectors the required fin force.

Since the shaft 34 is not rotating about the pitch axis θ; there is noleakage at the point where the shaft penetrates the waterproof membrane26. Furthermore, since the internal shaft 40 rotates internal to theshaft 34; there is no leakage. The pitch oscillation occurs within thehollow interior of the shaft 34; thereby, eliminating the possibility ofleakage. The elimination of leakage extends component life and theduration of a mission.

The diagram of the electromagnet 22 shown in FIG. 14 was used to modelthe force produced by each electromagnet. The magneto-motive force F isgiven by Equation (5):

$\begin{matrix}{F = \frac{({ni})^{2}\mu_{0}A}{2l_{gap}^{2}}} & (5)\end{matrix}$

wherein:

n=the number of turns in the coil in each electromagnet,

i=the current through the coil,

μ₀=the magnetic permeability in a vacuum,

A=the area, and

l_(gap)=the air gap between the magnet and steel in the direction of themagnetic flux.

Compared to μ₀, the magnetic permeability of cold rolled steel is 2000times larger. For a l_(gap)=3-4 mm at i=1.5 A, both Equation (5) andmeasurement of maximum force lead to a force of 0.74 N. In oneembodiment, the coils are formed by 266 turns of wire 23. The wire 23 ispreferably 26 AWG wire.

The hemispherical drive apparatus 10 actuates the flapping fin 42 forlow speed maneuvering and effects resonant oscillation of the fin.Resonant oscillation of the fin 42 is desired for several reasons: (a)displacement is amplified, (b) disturbances are rejected, (c) frictionis reduced and quality factor (Q) is increased (ratio of maximum kineticenergy to the total input energy), and (d) the motion of the shaft 34follows set parameters without the need for any sensors or anyconventional closed loop control. These properties are realizedmathematically by input/output circuitry 64 and are known asself-regulation. In accordance with the present invention,olivo-cerebellar dynamics are used to describe the mathematical form ofthe self-regulation. The model of the I^(th) ion-related controller isgiven by Equation (6):

$\begin{matrix}{\begin{bmatrix}\overset{\bullet}{z} \\{\overset{\bullet}{w}}_{i}\end{bmatrix} = {\begin{bmatrix}{{p_{iz}\left( z_{i} \right)} - w_{i}} \\{ɛ_{CA}\left( {z_{i} - I_{Ca}} \right)}\end{bmatrix} + {\begin{bmatrix}0 \\{- ɛ_{Ca}}\end{bmatrix}{I_{exti}(t)}}}} & (6)\end{matrix}$wherein:z_(i) and w_(i) are associated with sub-threshold oscillations andlow-threshold (Ca dependent) spiking;ε_(Ca) is a constant parameter that controls the oscillation time scale;andI_(Ca) drives the depolarization levels.

The nonlinear function is represented by Equation (7):p _(iz)(z _(i))=z _(i)(z _(i) −a _(i))(1−z _(i))  (7)The function I_(exti)(t) is the extracellular stimulus which is used forthe purpose of control (e.g., changing the motion of parameciumdirection). If the function I_(exti)(t)=0, then the nonlinear functionis given by Equation (8):p _(iz)(z _(i))=z _(i)(z _(i) −a _(i))(1−z _(i))  (8)wherein:a_(i) is a constant parameter associated with the nonlinear function.Equation (6) can be written as Equation (9) which represents theoscillation:{umlaut over (z)} _(i) +F(z _(i)){dot over (z)}+(kz _(i))+εI=0  (9)wherein:F is a cubic polynomial function; andk is a constant.The oscillator exhibits a closed-orbit Γ_(i) in the state space(z_(i)−ż_(i)); that is (z_(i)−w_(i)), which is also known as limit cycleoscillation (LCO), the constant parameters determining the form ofΓ_(i).Equation (2) is solved using the analog oscillators of input/outputcircuitry 64 (see FIG. 15).

The hemispherical drive apparatus 10 of the present invention providesmany benefits and advantages. The drive apparatus 10 is gearless andfrictionless and has a significantly lower noise level compared to priorart flapping fin propulsion systems. The drive apparatus 10 consumesrelatively less power than prior art systems, has a high quality factor(Q) and extends the duration of the mission. The drive apparatus 10provides roll oscillation in two planes; thereby, making the degree offreedom more similar to an animal. The drive apparatus 10 can be used toprovide a self-regulating trajectory to objects that require very smallforces (e.g., cilium-based nonlinear transducer). The drive apparatus 10is a modular design and also has the potential to reduce actuator drivesize.

The drive apparatus 10 can be adapted for use in orthopedicreconstruction. For example, the drive apparatus 10 can be developed asan elbow cap, knee cap or hip replacement. The hemispherical drive 10can also be used in the field of robotics.

The foregoing description of the preferred embodiments of the inventionhas been presented for purposes of illustration and description only. Itis not intended to be exhaustive nor to limit the invention to theprecise form disclosed; and obviously many modifications and variationsare possible in light of the above teaching. Such modifications andvariations that may be apparent to a person skilled in the art areintended to be included within the scope of this invention as defined bythe accompanying claims.

What is claimed is:
 1. A hemispherical drive apparatus comprising: ahemispherical housing having an interior and a cover; an array ofelectromagnets positioned within the interior of said housing, saidelectromagnetic array comprising a plurality of electromagnetics,wherein the location of each electromagnet is defined by correspondinglatitude and longitude coordinates; a gimbal positioned within saidcover of said hemispherical housing; a shaft engaged to said gimbal suchthat said shaft can roll oscillate in multiple planes, said shaft havinga first end that comprises ferromagnetic material and is located withinthe interior of said housing and a second end that is external to theinterior of said housing, the first end of said shaft being spaced apartfrom said array of electromagnets by a predetermined distance andremaining spaced apart from said array by the predetermined distance assaid shaft roll oscillates within the multiple planes; a fin located atthe second end of said shaft; and a control system to generate signalsthat sequentially excite predetermined electromagnets in accordance witha predetermined sequential order in order to produce an electromagnetictrack that is to be followed by the first end of said shaft; whereinsaid electromagnetic track forms a figure eight pattern that has acenter that is aligned with the center of said hemispherical housing. 2.The hemispherical drive apparatus according to claim 1, wherein themultiple planes comprises a first roll oscillation plane and a secondroll oscillation plane that is orthogonal to the first roll oscillationplane.
 3. The hemispherical drive apparatus according to claim 2,wherein said control system comprises a programmable processor havingstored therein the latitude and longitude coordinates of each saidelectromagnet.
 4. The hemispherical drive apparatus according to claim3, wherein said programmable processor is programmed with at least oneleast one sequence of latitude and longitude coordinates ofelectromagnets that are to be excited in sequential order in order toproduce the electromagnetic track.
 5. The hemispherical drive apparatusaccording to claim 1, wherein one pair of adjacent electromagnets areexcited at a time and each succeeding pair of excited electromagnets areadjacent to a pair of previously excited electromagnets.
 6. Thehemispherical drive apparatus according to claim 5, wherein saidhemispherical housing includes a cover disposed over an interior of saidhousing and a waterproof membrane disposed over said cover.
 7. Thehemispherical drive apparatus according to claim 6, wherein said shafthas a hollow interior and comprises an interior shaft rotatably disposedwithin the hollow interior, said internal shaft having a distal end thatis adjacent to the second end of said shaft, said fin being attached tothe distal end of said internal shaft.
 8. The hemispherical driveapparatus according to claim 7, further comprising an internal motorlocated within the hollow interior of said shaft, wherein said motor isengaged with said internal shaft and rotates said internal shaft aboutan axis.
 9. The hemispherical drive apparatus according to claim 8,wherein said control system is in electrical signal communication withsaid internal motor and outputs control signals to said internal motor.